I am reading John M Lee's Riemannian Manifolds : An Introduction to Curvature, which is very well written.
On page 16 : "Vector bundles are defined", quoting
A (smooth) $k$-dimensional vector bundle is a pair of smooth manifolds $E$ (total space) and $M$ (the base), together with the surjective map $\pi : E \to M $ (the projection) satisfying the following conditions : [...]
Because later defining smooth sections on page 19 : (quoting again)
If $\pi : E \rightarrow M $ is a vector bundle over M, a section of E is a map $ F : M \rightarrow E $ such that $ \pi \circ F = Id_M. $ [...]
So the point is : when one says $E$ is $TM$, i.e. the tangent bundle, then a point in E is $(p, T_pM)$ (in which case it will be a bijection) or $(p, V)$ where $V \in T_pM$ ?
I am confused between these two concepts : especially "section".
Some reference or explanation with a clear example will be really helpful.